4 Climate Dynamics¶
4.1: Approximation¶
For this, we follow the simplified climate dynamics used in \mycitet{BarnettBrockHansen:2021}. Their approach is based on an approximation from the geoscience literature used to support model comparisons. Specifically, \cite{MatthewsGillettScott:2009} and others have purposefully constructed an approximation for climate models outputs:
where TCRE is an acronym for the Transient Climate Response to cumulative Emissions. This simplified formulation abstracts from transitory ``weather’’ fluctuations in temperature. Instead emissions today have a long-lasting impact on temperature in the future where TCRE is a measure of climate sensitivity.
Our specific form:
for \(\theta(\ell) \in \Theta\) where \(\theta(\ell)\) is a specific TCRE obtained from the set of TCRE models \(\Theta\). Here we constructed this histogram. For baseline probabilities, we presume that each model has subjective probability \({\frac 1 {L_y}}\) and \(L_y\) is the number of climate models we use as inputs. In other words, we treat each such model as having the same subjective probability. We abstract from parameter learning since learning about such parameters has been very slow.
4.2 Modeling Heterogeneity¶
More specifically, we use exponentially weighted average of each of response functions as coefficients \(\theta(\ell) \in \Theta\) in our computations, where the discount rate is set as \(\delta=0.01\) and the number of climate models is \(L_y = \# \Theta = 144\).
The histogram of these coefficients are represented below:
[1]:
from src.plot import plot_hist
plot_hist("""Figure 1: Histograms for climate sensitivity parameters""")
4.3 Intensity function¶
\begin{equation} {\mathcal J}_n(y) = \left\{ \begin{matrix} {\sf r}_1 \left( \exp \left[ {\frac {{\sf r}_2} 2} (y -{\underline y})^2\right] - 1 \right) & y \ge {\underline y} \cr 0 & 0 \le y < {\underline y} \end{matrix} \right. \end{equation}
Where the values for \(r_1\) and \(r_2\) are as follows:
Parameter |
Value |
|---|---|
\(r_1\) |
1.5 |
\(r_2\) |
2.5 |
[2]:
from src.plot import plot_intensity
plot_intensity("""Figure 2: Plot of the Intensity Function""")
4.4 Climate Damages¶
We assume that capital, output, both investments, and consumption are all diminished proportionately by \(N_t\). Our damage specification uses a piecewise log quadratic specification as a function of the temperature anomaly \(y.\) We suppose that the derivative of the logarithm of damages \(\hat{n}\) with respect to the temperature anomaly is
\begin{equation} \begin{array} {llll} \frac {d {\hat n}} {dy} & = \lambda_1 + \lambda_2 y & y \le {\tilde y} \\ \frac {d {\hat n}} {d{y}} & = \lambda_1 + \lambda_2 \left( y - {\tilde y} + {\bar y} \right) + \lambda_3(z_n ) (y - {\tilde y}) & y > {\tilde y} \end{array} \end{equation} for \(z_n \in \left\{ 1, 2, ..., L_n \right\}\).
This equation has an initial condition \({\hat n}(0) = 0.\) In the stochastic version of what follows, \({\tilde y}\) will be triggered by a Poisson jump prior to a temperature threshold \({\bar y}.\) We specify the intensity so that this jump takes place in the interval \([{\underline y}, {\bar y}].\) We shift the derivative of damages with respect to temperature to the right as captured by the change from $:nbsphinx-math:lambda_2 y $ to $:nbsphinx-math:lambda_2 {\bar `y} $. We also increase the slope by including a term :math:lambda_3(z_n) ( y - {tilde y})`, where the coefficient \(\lambda_3(z_n)\) is ex ante uncertain.
We plot the implied damage functions for thresholds between \({\tilde y} = 1.5\) and \({\tilde y} = 2.\), including a range of \(\lambda_3\)’s used in our quantitative policy assessment.
[ ]:
from src.plot import plot_damage
plot_damage("""Figure 3: Range of Possible Damage Functions for Different Jump Thresholds""")